Application Local Polynomial and Non-polynomial Splines of the third order of approximation for the construction of the Numerical Solution of the Volterra Integral Equation of the Second Kind I.G.BUROVA Department of Computational Mathematics St. Petersburg State University 7/9 Universitetskaya nab., St.Petersburg, 199034 RUSSIA Abstract: - The present paper is devoted to the application of local polynomial and non-polynomial interpolation splines of the third order of approximation for the numerical solution of the Volterra integral equation of the second kind. Computational schemes based on the use of the splines include the ability to calculate the integrals over the kernel multiplied by the basis function which are present in the computational methods. The application of polynomial and nonpolynomial splines to the solution of nonlinear Volterra integral equations is also discussed. The results of the numerical experiments are presented. Key-Words: - Volterra integral equation, non-polynomial spline, polynomial spline Received: December 21, 2019. Revised: January 25, 2021. Accepted: February 19, 2021. Published: March 2, 2021. 1 Introduction Here we study the application of the local polynomial and non-polynomial interpolation splines of the third order of approximation for the construction the numerical scheme for the solution of the Volterra integral equation of the second kind. There are many numerical methods for solving Volterra integral equations of the second kind. The most common numerical methods are based on the use of quadrature formulas. In connection with the emerging needs for constructing methods of high accuracy, many researchers, again, resort to modernizing the known methods for solving integral equations and construction the new ones. The authors of papers [1]-[10] devoted a lot of attention to the modification of the known numerical methods and the construction of new numerical methods for solving integral equations. In study [1] a numerical scheme for approximating the solutions of the nonlinear system of fractional-order Volterra- Fredholm integral differential equations was proposed. The main characteristic of this approach is that it reduces such problems to a linear system of algebraic equations. In paper [2], a new and efficient method for solving the three-dimensional Volterra-Fredholm integral equations of the second kind, first kind and even singular type of these equations is presented. Here, the authors discuss three variable Bernstein polynomials and their properties. This method has several advantages in reducing the computational burden with a good degree of accuracy. Furthermore, the authors obtain an error bound for this method. A computational technique based on a special family of the MΡuntz-Legendre polynomials to solve a class of Volterra-Fredholm integral equations is presented in paper [3]. The proposed method reduces the integral equation into algebraic equations via the Chebyshev-Gauss-Lobatto points, so that the system matrix coefficients are obtained by the least squares approximation method. The useful properties of the Jacobi polynomials are exploited to analysis the approximation error. Spline functions were used to propose a new scheme for solving the linear VolterraβFredholm integral equations of the second kind in paper [4]. Two types of non-polynomial spline functions (linear, and quadratic) were used in paper [5] to find WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2021.20.2 I. G. Burova E-ISSN: 2224-2880 9 Volume 20, 2021
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Application Local Polynomial and Non-polynomial Splines of the third
order of approximation for the construction of the Numerical Solution
of the Volterra Integral Equation of the Second Kind
I.G.BUROVA
Department of Computational Mathematics
St. Petersburg State University
7/9 Universitetskaya nab., St.Petersburg, 199034
RUSSIA
Abstract: - The present paper is devoted to the application of local polynomial and non-polynomial
interpolation splines of the third order of approximation for the numerical solution of the Volterra integral
equation of the second kind. Computational schemes based on the use of the splines include the ability to
calculate the integrals over the kernel multiplied by the basis function which are present in the computational
methods. The application of polynomial and nonpolynomial splines to the solution of nonlinear Volterra
integral equations is also discussed. The results of the numerical experiments are presented.
Key-Words: - Volterra integral equation, non-polynomial spline, polynomial spline
Received: December 21, 2019. Revised: January 25, 2021. Accepted: February 19, 2021.
Published: March 2, 2021.
1 Introduction Here we study the application of the local
polynomial and non-polynomial interpolation
splines of the third order of approximation for the
construction the numerical scheme for the solution
of the Volterra integral equation of the second kind.
There are many numerical methods for solving
Volterra integral equations of the second kind. The
most common numerical methods are based on the
use of quadrature formulas. In connection with the emerging needs for
constructing methods of high accuracy, many
researchers, again, resort to modernizing the known
methods for solving integral equations and
construction the new ones. The authors of papers
[1]-[10] devoted a lot of attention to the
modification of the known numerical methods and
the construction of new numerical methods for
solving integral equations. In study [1] a numerical
scheme for approximating the solutions of the
nonlinear system of fractional-order Volterra-
Fredholm integral differential equations was
proposed. The main characteristic of this approach
is that it reduces such problems to a linear system of
algebraic equations.
In paper [2], a new and efficient method for solving
the three-dimensional Volterra-Fredholm integral
equations of the second kind, first kind and even
singular type of these equations is presented. Here,
the authors discuss three variable Bernstein
polynomials and their properties. This method has
several advantages in reducing the computational
burden with a good degree of accuracy.
Furthermore, the authors obtain an error bound for
this method. A computational technique based on a
special family of the MΡuntz-Legendre polynomials
to solve a class of Volterra-Fredholm integral
equations is presented in paper [3]. The proposed
method reduces the integral equation into algebraic
equations via the Chebyshev-Gauss-Lobatto points,
so that the system matrix coefficients are obtained
by the least squares approximation method. The
useful properties of the Jacobi polynomials are
exploited to analysis the approximation error.
Spline functions were used to propose a new
scheme for solving the linear VolterraβFredholm
integral equations of the second kind in paper [4].
Two types of non-polynomial spline functions
(linear, and quadratic) were used in paper [5] to find
WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2021.20.2 I. G. Burova
E-ISSN: 2224-2880 9 Volume 20, 2021
the approximate solution of Volterra integroβ
differential equations.
A computational method for solving nonlinear
Volterra-Fredholm Hammerstein integral equations
is proposed in [6], by using compactly supported
semiorthogonal cubic B-spline wavelets as basis
functions. The non-polynomial spline basis and
Quasi-linearization method to solve the nonlinear
Volterra integral equation were used in [7]. A new
collocation technique for the numerical solution of
the Fredholm, Volterra and mixed Volterra-
Fredholm integral equations of the second kind is
introduced in [8], and a numerical integration
formula on the basis of the linear Legendre multi-
wavelets is also developed in [8].
Note that in papers [4]-[10] splines are used to
construct new numerical methods. The construction
of various splines and wavelet splines is considered
in papers [11]-[16]. The approximations with
splines on the irregular set of nodes are of particular
interest [13].
The application of the generalized Haar spaces is
sometimes very useful [14]. Paper [15] deals with
the use of the first two vanishing moments for
constructing cubic spline-wavelets orthogonal to
polynomials of the first degree. The method
proposed in [16] can be used to calculate the real
eigenvalues of an arbitrary matrix with real
elements. This method uses splines of the
Lagrangian type of the fifth order and/or polynomial
integro-differential splines of the fifth order.
In paper [17] the application of the finite-difference
methods are investigated to compute the definite
integrals.
At present, the theory of approximation by local
interpolation splines continues to evolve.
Approximation with local polynomial and local non-
polynomial splines of the Lagrange types can be
used in many applications. Approximation with the
use of these splines is constructed on each mesh
interval separately as a linear combination of the
products of the values of the function at the grid
nodes and basic functions. The basis functions are
defined as a solution of a system of linear algebraic
equations (approximation relations). The
approximation relations are formed from the
conditions of accuracy of approximation on the
functions forming the Chebyshev system. The
constructed basic splines provide an approximation
of the prescribed order which is equal to the number
of equations in the system, or, in other words, it is
equal to the number of grid intervals in the support
of the basic splines. Using the basic splines, we can
construct continuous types of approximation [10]-
[12], [16]. This paper continues the construction of
numerical methods based on the use of spline
approximations [10]. The proposed numerical
methods extend the set of known numerical methods
for solving integral equations [18].
The paper is organized as follows. Section 2
discusses the theoretical aspects of the application
of polynomial and non-polynomial splines of the
second order of approximation. Section 3 considers
the properties of polynomial and non-polynomial
splines of the third order of approximation. A
numerical method for solving the Volterra equation
of the second kind is also proposed here. Section 4
presents the results of the numerical solution of the
Volterra equations of the second kind using the
trapezoidal method, using polynomial and non-
polynomial splines of the second order of
approximation, as well as using splines of the third
order of approximation.
2 Application of Splines of the
Second Order of Approximation In paper [10] the numerical solution of Volterra-
Fredholm integral equations of the second kind was
constructed with the use of local splines of the
second order of approximation.
As it is shown in paper [11], if the functions π1,
π2 form a Chebyshev system, then the basis
functions ππ , π = π, π + 1, can be determined by
β = 0.1 on [β1,1]. Tables 1, 2 show the actual
errors of approximation of some functions obtained
with the use of the polynomial and non-polynomial
splines of the third order of approximation. Table 3
shows the actual errors of approximation of some
functions obtained with the use of the polynomial
and non-polynomial splines of the second order of
approximation.
Table 1. The actual errors of approximation of some
functions obtained with the use of the polynomial and
non-polynomial splines of the third order of
approximation
π’(π₯) π1(π₯) = 1,
π2(π₯) = π₯,
π2(π₯) = π₯2
π1(π₯) = 1
π2(π₯)
= cos(π₯), π3(π₯)= sin(π₯)
π1(π₯) = 1,
π2(π₯)
= exp(βπ₯),
π3(π₯)
= exp (π₯)
exp(π₯) 0.000160 0.000320 0.0
π in(π₯) 0.0000641 0.0 0.000128
π₯2 0.0 0.000127 0.000126
exp(βπ₯) 0.000172 0.000344 0.0
π in(2π₯) 0.000512 0.000384 0.000639 1
1 + 25π₯2
0.0296 0.0294 0.0297
Table 2. The actual errors of approximation of some
functions obtained with the use of the polynomial and
non-polynomial splines of the third order of
approximation
π’(π₯) π1(π₯) = 1,
π2(π₯)
= exp(π₯),
π3(π₯)
= exp (2π₯ )
π1(π₯) = 1,
π2(π₯)
= exp(βπ₯),
π3(π₯)
= exp (β2π₯)
exp(π₯) 0.0 0.0
π in(π₯) 0.000200 0.000187
π₯2 0.000666 0.000593
exp(βπ₯) 0.00108 0.0
π in(2π₯) 0.000843 0.000775 1
1 + 25π₯2
0.0272 0.0311
Table 3. The actual errors of approximation of some
functions obtained with the use of the polynomial and
non-polynomial splines of the second order of
approximation
π’(π₯) π1(π₯) = 1,
π2(π₯) = π₯.
π1(π₯)
= cos(π₯), π2(π₯)= sin(π₯)
π1(π₯) = 1,
π2(π₯)
= exp (βπ₯)
exp(π₯) 0.00323 0.00647 0.00646
π in(π₯) 0.00102 0.0 0.00177
π₯2 0.00250 0.00363 0.00487
exp(βπ₯) 0.00323 0.00647 0.0
π in(2π₯) 0.00498 0.00374 0.00558 1
1 + 25π₯2
0.0418 0.0407 0.0443
Consider the approximation by polynomial splines.
Let the second and third derivatives of the function
WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2021.20.2 I. G. Burova
E-ISSN: 2224-2880 15 Volume 20, 2021
π’(π₯) be such that max[π,π]
|π’β²β²β²| β€ πΆ, max[π,π]
|π’β²β²| β€
πΆ, πΆ = ππππ π‘ > 0. The example is function
π’(π₯) = exp(π₯)The calculation results presented in
Tables 1-3 confirm the fact, that the splines of the
third order of approximation in this case give a
smaller error than the splines of the second order of
approximation if in both cases we apply the same
step β. A similar statement can be formulated for
non-polynomial splines. The theorems formulated
above give asymptotic estimates. This allows us to
hope that for sufficiently small β, the use of splines
of the third order of approximation will give a
smaller error than the use of splines of the second
order of approximation.
Π‘onsider the following example. We will
approximate the Runge function π’(π₯) =1
1+25π₯2 on
a uniform grid of nodes built on the interval [π, π] =[β1,1] with the step β . Fig. 1 shows a graph of the
absolute value of the second derivative of the Runge
function. Fig. 2 shows a graph of the absolute value
of the third derivative of the Runge function. We
use the results of Theorem 2 and Theorem 4.
Let us introduce the notations:
π΄(β) = β2/8 max[0,1]
|π’β²β²(π₯)|,
π΅(β) = 0.0625β3 max[0,1]
|π’β²β²β²(π₯)|.
The plots of π΄(β) (blue), and π΅(β) (red) are given
in Fig.3.
Solving the equation π΄(β) = π΅(β), we find β0 β0.171. At this point, the graph lines intersect. When
β is greater than this value (β > β0), the theoretical
error when using polynomial splines of the second
order of approximation will be lesser then when
using polynomial splines of the third order of
approximation. It is easy to calculate that when β =0.3 we get π΄(β) β 0.985, π΅(β) β 0.562. Let β = 1/3. Fig. 4 shows plot of the actual error of
approximation of the Runge function by splines of
the second order of approximation. The maximum
of the absolute value of the actual error is 0.0623.
Fig. 5 shows plot of the actual error of
approximation of the Runge function by splines of
the third order of approximation. The maximum of
the absolute value of the actual error is 0.236.
Fig.1. The plot of graph of the absolute value of the
second derivative of the Runge function.
Fig.2. The plot of the graph of the absolute value of the
third derivative of the Runge function
Fig.3. The plots of π΄(β) (blue), and π΅(β) (red).
Fig.4. The plot of the actual error of approximation of the
Runge function by splines of the second order of
approximation
Fig.5. The plot of the actual error of approximation of the
Runge function by splines of the third order of
approximation
Thus, there are cases when linear polynomial splines
will give a smaller approximation error than
quadratic ones. therefore, to verify the result, both
types of approximations should be applied.
Now let's apply splines to the calculation of the
integral β« πΎ(π₯, π )π’(π )ππ π₯
π.
Transforming the integral β« πΎ(π₯, π )π’(π )ππ π₯π+1
π₯π ,
π = 1, β¦ π β 1, using formula (3), we obtain
WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2021.20.2 I. G. Burova